 So welcome everyone. I think we all know why we're here. This is a special gathering for Peter's 70th birthday. I remember when Peter turned 60, which is Babylonian 10, we had to travel to Israel and then to China and then to Princeton. So now we can all sit from the comfort of our homes and enjoy some nice reminiscences and conversations and show our appreciation to Peter for all that he's done for all of us. I'm sure I don't speak for just myself. So for this honor, we're very glad to have Professor Zev Rudnik from Tel Aviv University. He is the beer chair in number theory at Tel Aviv. He received his PhD from Yale and was a postdoc at Zego Assistant Professor at Stanford and then was an Assistant Professor at Princeton before moving to Tel Aviv and then many other honors and awards. But I won't go on with that today. We'll just hear from Zev and please Zev take it away. Okay, thank you Alex. So it's a pleasure to speak on Peter's 70th birthday. I had a vague plan to make a retrospective all of Peter's work in the first 70 years and then realized very quickly that it was a foolish idea and I'm going to concentrate on some aspects of his work on automorphic forms and only on classical automorphic forms by which I mean either classical modular forms which I will I do think most everyone knows what they are and then the more exotic mass waveforms and I feel that I should recall the definition of a modular form even though I expect everyone to know it and then while thinking of how to define it I said let's have Peter do that so I really can't explain what a modular function is in one sentence I can try and give you a few sentences to explain. I really can't do it in one sentence. Oh it's impossible. Okay that wasn't very good so let's try let's try to recall the definition. So a classical modular form is a holomorphic function on the aft behalf plane which satisfies a lot of functional equation under the modular group or equivalently it's periodic and then is transforms in a certain way under the Mavius reflection so these are two of the conditions and there's a third one which is a little more complicated to explain which is holomorphic at infinity and what that means is you map the aft behalf plane to the punctured unit disk by taking tau the upper half plane variable to e to the 2 pi i tau which is called q the known and then the periodicity condition on the function implies among other things that the modular form is only a function depends only on q on this new variable and that as a function of q it's analytic in the image of the upper half plane so in the punctured unit disk and holomorphic at infinity means that we require this new function to have an analytic continuation at the origin and then if you look at the Taylor series of this new function of q this is called the Fourier expansion of the modular form this is what a lot of time encodes the important arithmetic information in the shape of the coefficients these Fourier coefficients then the last definition is what it means to be cuspid it means that this new function of q vanishes at the origin equivalently equivalently that the period of the modular form is zero so that's the definition of modular form and classical cuspom so i'll denote by m of k the space of modular form of weight k and inside it is the subspace of cuspoms of co-dimension one typically and modular forms have a topological origin and the dimension grows linearly with k the dimension grows like k over 12 plus a bounded quantity so the classical examples are eisenstein series so you just take something and periodicize it and then by definition essentially it's a modular form of weight k and then the Fourier expansion of the eisenstein series is clearly of arithmetic origin the coefficients are essentially divisor sums so it's a non-cuspital modular form of weight k and then you can write the space of modular forms as multiples of the eisenstein series plus cuspoms another example is a modular discriminant it's the unique cuspom of weight 12 it can be defined by this infinite product which clearly has an integer Fourier expansion from this form it's not at all obvious that it is a modular form of weight 12 but it is and then another important source of examples are so-called Hecke eigenforms and one way of defining them is saying that the Fourier coefficients are multiplicative in fact satisfies slightly more stringent requirements that it's called the Hecke relations which are written down here but when m and m are co-prime it reduces to multiplicativity so for instance the modular discriminant is a Hecke eigenforms the coefficients are multiplicative in this sense again this is not at all obvious to me and another example is the eisenstein series the the divisor sums satisfy this Hecke relation so these are classical modular forms I think everyone knows them there's another class of automorphic forms of classical automorphic forms called mass cuspoms and these I think were very obscure to most number theorists until Peter started proselytizing for them Peter among other people but he was really instrumental in this so what's the definition so we again look at functions on the upper half plane but instead of being holomorphic we require them to be eigenfunctions of the laplacian of the hyperbolic laplacian so that in particular implies that they are real analytic and they are modular forms of weight zero so they they are invariant and they may be transformations for the modular group and then cuspid means that the zero's Fourier coefficients vanishes and they have a Fourier expansion which is similar to the q expansion of classical modular forms except instead of the function e to the 2 pi i tau you get something a little more exotic in the y variable with the k bessel function in there whose parameter is related to the Laplace eigenvalue now I wrote down a definition but it's not it's not really easy to write down examples of this definitions and I said that homomorphic modular forms exist for topological reason I said that the space of modular forms as dimension which grows linearly with the k existence of mass forms is a more delicate question so I wrote down I wrote down a definition but it's not clear that I can produce examples now for congruence certain congruencer groups of the modular group who already mass constructed such mass cusp forms but that construction doesn't work for the full modular group and it is only thanks to selberg that we know that these exist selberg showed that mass cusp forms exist by using a counting argument which is called vile's law they approved vile's law for for mass mass cusp forms on s2z and the same and likewise for the existence of mass cusp form for any congruence subgroups and the the reasons that they can prove existence are arithmetic in nature in particular that is invented is a famous selberg trace online part for this reason and to establish the existence of a mass form that you need arithmetic information particularly the connection to the theory of l functions and then some information about the distribution of zeros of l functions the original functions anyway this this is just to say that existence of mass cusp form for s2z is a fact but it's not a trivial fact and they can't write down examples in a simple way I'm going to return to this theme in the minute now if you replace the modular group by some other discrete subgroup which has a non compact fundamental domain but is not arithmetic it is not known if there are any mass cusp forms and if there are how many there are so that that counting argument the vile that alluded to is not known though so for instance if we look at what are called the hecke groups so there are groups generated by the standard reflection and by a translation not by one but by twice cosine two pi over q q is an integer so here is the fundamental half of the fundamental domain for this hecke group so it has it's a hyperbolic triangle with base angles pi over two pi over q and zero and these are non-arithmetic groups I haven't defined what arithmetic means unless q is three four six when q is three you get the modular group so for q equals three we know that the mass cusp forms I said that selberg pulled the existence but if q is not one of these three numbers we don't know that and it was believed that mass cusp forms had to exist it was it was called the raucous selberg conjecture and then at some point it became raucous conjecture and one of the fundamental insights that peter had in his work with ryle phillips is that they probably don't exist unless there's an arithmetic reason so following works by a series of work by ryle phillips and peter saunak and then by scott walpert and when she lived and very recently relatively recently by look here and chris judge it's now reasonable to believe that for non-arithmetic groups few mass cusp forms exist I actually don't quite understand the exact formulation of that so people peter is allowed to interject and give an exact formulation if he can drag himself off the beach in the caribbeans to do that in any case I think I'll skip I'll skip I'll skip I'm slipping okay so the challenge is to to prove to prove this so um chris judge and lucy liaret proved the toy case of this that peter challenged them too but as far as I know uh this what I stated this conjecture is still open so for the for these hecke triangle groups whether they are whether there exist mass cusp forms okay so I was saying that mass cusp forms are mysterious objects and I can't write them down and uh there's for me a very um beautiful uh insight of peter's of how to ask this in the precise form and the the the question in this in this special context of the modular group asks can I write down the mass cusp form which has integer coefficients like the modular discriminant or the eisenstein series and what peter proved is that you cannot do that so there are no hecke mass cusp forms so there are no mass cusp forms with multiplicative coefficients for s o to z whose coefficients are integers now if so I think part of the 50 percent of the work here is to think to ask this question and then if you know what the langen program predicts then the answer is clearly no that cannot be any the other hand uh we don't have the langen program so we don't know the romanian conjecture for these and we don't know the satiate conjectures for these mass forms but peter was able to use substitutes for these to to deduce what I I wrote down here in a very special case okay so now I want to move to a different question on mass forms which is uh uh peter's work on quantum unique egodicity so this is something that we originally formulated also long ago in a very general context for eigenfunctions of the population on any negatively curved let's say compact manifold but it makes sense to ask this for mass cusp forms so now the the correct formulation of quantum unique egodicity is more complicated than I want to to give here but the poor man's version is the following so you take your a sequence of mass past forms for the full modular group so the eigenfunctions of the population and take a sequence where the eigenvalue goes to infinity now selberg guarantees for us that there are such there is such a sequence and then the quantum unique egodicity conjecture is that the measure is defined by by taking the density against hyperbolic measure as the square of this mass form so that these measures converge to the hyperbolic measure as the plus eigenvalue grows to infinity so this is the quantum unique egodicity conjecture I should emphasize that it is still open it's still open and I want to explain a little bit more about this QE conjecture by specializing it to a more classical subject which is a version of this for holomorphic modular forms and and then some elementary applications okay so the version for holomorphic modular forms is very similar so you take a sequence okay so now it's a definition it's not a conjecture so the definition is you take a sequence of holomorphic cusp forms of increasing weight and you say that it satisfies QUE quantum unique egodicity if for every compact subset of the fundamental domain the measure is defined by taking the square of this holomorphic cusp form converge to the hyperbolic measure or equivalently what this is what I said the you test it against your favorite nice test function which will follow along to the constant and these periods should tend to zero so this is the formulation of holomorphic quantum unique egodicity now it's a definition it's not a conjecture because it depends on the sequence so some sequences will satisfy may satisfy this and some may not and because the space of holomorphic cusp forms is very big as I said its dimensionality grows linearly with k there are certainly subsequences which we can write down which violate QUE but there are some special interesting sequences where QUE does hold and a very important case is of heke eigen forms so this was proved by Holovinsky and sound about 15 years ago or so even longer right I'm not responding to the chat because it's hard to see but if people have questions they're welcome to interject I actually prefer listening to questions and to lecture so let me describe what goes into QUE for heke holomorphic heke eigen forms so what is what was holomorphic QUE you take and now your favorite test function and consider this its integral against these measures defined by the squares of the cusp forms and you want this integral to go to zero as long as the test function is orthogonal to the constants now in the case when the test function and the cusp forms are heke eigen forms and they're in their respective domains there's a beautiful formula that was envisioned by Peter and was proved by Peter student Tom Watson which relates the square of this periods to the special value of an l function to the value of a degree six a gl six l function in the center of the critical strip so it says that the square is one over k which is great we want this to go to zero so one over k is this is fantastic times the special the central value of this exotic l function so in order to prove QUE what we all we need to know is that this central value grows slower than k k is the parameter of the module of form the cusp form which appears here see the test function is fixed the only growing thing is the weight of the cusp form of the cusp form now if we look at what we know from the theory of l functions there's a classical bound that you can get for the central values which is called the convexity bound and it says roughly so i'm lying through my teeth here but not not in a serious way it says roughly that the central value of this particular family of l function is bounded by k and we are dividing by k so we just miss tending to zero on the other hand the ringman hypothesis implies what's called the lindel of hypothesis which says that this central value is essentially bounded as a function of k and then we're in great shape this will say that this the square of the period is essentially one over k and so certainly goes to zero so once you have watson's formula you know that the ringman hypothesis implies qe now you don't really need the ringman hypothesis you need something much less you just need something a growth that the growth of the central value is smaller than k let's say k to the one minus delta so this is called sub convexity and unfortunately it's still unknown sub convexity in the sense that i have described of getting a a bound which is k to a power lesson one is still unknown in this case anyway this this problem and other problems spurred a huge burst of activity to establish sub convexity for various families of l functions and it's still an ongoing activity but unfortunately none of no one has been able to establish sub convexity for this particular family nonetheless i was saying that qe for this family was proved by holovinsky and sound because they did slightly less than sub convexity but still enough to show that this tends to zero again i'm lying through my teeth here there's all kinds of other factors here which don't allow me to say exactly what what you proved but anyway they they proved that this period goes to zero and this is a later development with qe the original qe for hecke mass forms was proved by long linens first and for groups through compact fundamental domain and then sounded it for let's say the modular group now so this was a technical slide holomorphic qe is a little technical to to formulate but it has implications to a much more classical subject in the theory of modular forms which is the distribution of zeros and this is what i now want to spend the rest of the talk about and hopefully finish much earlier than the the hour okay so um so the basic fact about the modular form of weight k is that it has lots of zeros in the fundamental domain it's a holomorphic function and it has about k over 12 zeros so in fact the number of zeros if you count them with some multiplicities is exactly the dimension of the space of cuss forms of weight k so the question is where are they located and they want to discuss the location of these zeros for a number of families of modular forms and so i'll start with eisenstein series and then describe three more families depending on how much time i have left i think i have plenty of time so the first work on this subject was done in the 60s so at that time there was people were looking at zeros of this eisenstein series they described and then the people found the zeros of first of the first eisenstein series of weight roughly up to 40 so Robert Rankin was very active in this and then he conjectured that for this eisenstein series all the zeros lie on this arc on the boundary arc of the fundamental domain i'm not including the this the right half of this arc because it's it's equivalent under the modular group to the left half of the arc so he conjectured that all zeros lie there and then probably within a few days after seeing this or a few minutes uh Swinnett and Dyer understood how to prove it so he and not Robert Rankin but Fenny Rankin Rankin's daughter have a very short paper proving that the zeros of eisenstein series all lie on this arc and in fact they are uniformly distributed very rigidly spaced here so this is one example of will of the location of zeros of a family of modular forms eisenstein series and there are lots of works of taking this rankin Swinnett and Dyer method and trying to apply to other families of modular forms here's another family um Poincare series so Poincare series uh defined in a very similar way to eisenstein series you take something and sum it over the lattice over the modular group uh there's a typo here um and uh so you take uh k which is bigger than two an even integer and some positive integer m and form the series when m is zero we get our eisenstein series when m is bigger than zero you get a new thing which turns out to be a cusp form not an eisenstein series it's a cusp form and in fact as you vary m for a fixed k you get a basis of the space of cusp forms of weight k so I remind you the the dimension of the space of cusp form of weight k is k over 12 essentially so Rankin um went back to the subject in 82 and then examined the distribution of zeros of this Poincare series and he proved that at least some of these zeros lie on this um arc on the bottom of the fundamental domain so precisely at least k over 12 minus m of the zeros lie on this arc so if m is fixed and k is growing then it says essentially all the zeros almost all the zeros lie on the arc um but then uh very recently like a few days ago Noam Kemal who is a PhD student at Tel Aviv asked what happens if you take m to be growing with k growing linearly with k and then what Noam discovered is that something completely different happens a positive proportion of the zeros actually lie on the other part of the boundary of the fundamental domain and then depending on the ratio between m and k there will also be zeros on the imaginary axis and what's special about these lines so the this line the imaginary arc axis and the boundary of the fundamental domain for all of them the j invariant is a real number so these are called real zero so this is what happens for another family of holomorphic modular forms we had eisenstein series now we have Poincare series which are very similar but exhibit a different behavior depending on the the range of parameters at hand now perhaps all right sorry is that proof the previous theorem is proved by counting like like swinit and dyre and k um it's it's a little different than swinit and dyre um again Noam the idea of swinit and dyre is to take this series and say the one or two important terms and the rest are very small and the important terms uh give you something that oscillates and you can count the oscillation so from that point of view it's it's the same idea but here the oscillations instead of taking part on the boundary on the on the arc they take part either on the right boundary or on the imaginary axis depending on the range of parameters so it's the same spirit of idea yeah thanks good um and now uh a much more complicated example is uh hecke eigenforms castle the hecke eigenforms and then uh something completely different happens so instead of the zeros being localized on a one-dimensional set that you can put your hand on like the bottom of the boundary the the bottom of the fundamental domain or the imaginary axis or the or the other boundary of the fundamental domain um it turns out that as you increase the weight the zeros become dense and not only do they become dense they become uniformly distributed with respect to the hyperbolic measure so the the theorem is as i stated here you take a a sequence of hospital hecke eigenforms of growing weight and then um if you fix any nice subset compact subset of the fundamental domain then the zeros of each guy the each each guy of weight k has about k over 12 zeros um the proportion of zeros which lie in this compact subset tends to the relative area of that compact subset so in particular that dense now this is something that i realized it was true about 25 years ago following some developments in quantum chaos and then i waited patiently until Watson's formula was written down because once Watson's formula was written down um we knew that holomorphic u u e follows from g r h and um what one shows is that holomorphic u u e quantum unique to go this implies this equidistribution statement so once Watson's formula uh i can't say was published but was written down uh we knew that this equidistribution is true at least pieces and it's on the or go yeah i know i see but i said it was never yeah right and in any case i was telling you that holovinsky and sound actually proved holomorphic q u without uh summing g r h so this is actually a a a a theorem now now there's a question in the chat about q e for eisenstein series i don't know if you want to answer yeah would i expect three four holomorphic eisenstein series so the answer is definitely not yes you just answered the question thanks because so this is one way so for the two families i showed earlier holomorphic eisenstein series or holomorphic poincare series here is the proof that they don't satisfy q e i'm not saying it's the best proof but it is a proof because we saw the zeros don't become dense they accumulate on a one-dimensional set so we now for free once we have this for free we we know that they don't satisfy q e well what about poincare series for associative geodesics in the modular surface it's a good question i don't have the answer that's a great question that would be a very nice thing to investigate what uh where do they the zeros lie that would be a great question thank you very surprised if the zeros become dense but i don't know the answer if anyone has a clue then uh shout out or we can discuss this in a few minutes if someone is right so the question is if you take poincare series as sort of to a closed geodesic i'm not i'm not defining them what can you say about the zeros good right so uh peter and amit gauche went back so this is a theorem right the mathematics it's kind of boring once you have a theorem it's very hard to argue with this unlike physics but they said went back to this and um they asked okay what happens not in a fixed compact subset of the random element in the plane what if you go up towards the cusp what happens to the zeros now here is a picture of the zeros of a particular hecke eigen form of way 2000 by feathery stormberg so you know this this thing has um about 180 zeros also here's a picture of them and you see that the zeros are sort of dense at least in the bottom of the fundamental domain but if you look carefully you also see a few zeros lying on the boundary of the fundamental domain and also a few zeros smack on the imaginary axis and that's kind of surprising because if the zeros dense you should think that there's zero probability of them lying on your favorite straight line when you're fairly geodesic but nonetheless in the picture you see this and then by building a random model so people who know random polynomials know that they actually have real zeros surprisingly they built a random model which predicts that there are actually not so few zeros which lie on these boundary geodesic and on the imaginary axis so as we said there are real zeros because the j invariant is real and so they conjectured that the number of real zeros is about root k log k um there's a precise constant in the and that once you go sufficiently high up in the cusp at height slightly bigger than root k then almost all of the zeros actually are confined to these two geodesics and this conjecture as far as I know is still open peter and amit prove that a lot of zeros actually lie on these vertical geodesics but not k to the half but some smaller l of k and kaisa matomaki uh improved this exponent but as far as I know this conjecture which I think is really beautiful is still open and people are welcome to correct me right so uh we have three families so far of modular forms where we understand the distribution of zeros this is eisenstein series von karei series and then hecke eigen forms and with hecke eigen forms there's further questions that you can ask which are still open right um and the last example is again very recent it's from this summer so I looked at cusp forms which vanished to very high order at the cusp so hecke eigen forms vanished to all the one of the cusp the q expansion starts with q but uh any cusp any modular form first of all cannot vanish to infinite at infinity to order higher than the dimension of the space of cusp forms so k over 12 roughly and it's determined by its first um l plus one coefficients l being this dimension so here's an example of a cusp form you take a form which all of its initial coefficients are zero except one and let's say you take d equals four and you look at form which a form which looks like q to the l minus four plus everything zero up to the coefficient of q to the l plus one that uniquely determines the unique unique such form okay and such a form will have exactly d these four let's say zeros in the interior of the fundamental domain all the other zeros lie in the cusp it says the high order vanishing infinity and then I ask what can I say about these four zeros they're not going to be dense anyway right there's four points they're not going to be dense um so here is the result let's look at the picture so if people can see the colors the the color blue here there are four zeros so this is a particular example here of weight um 1000 so that's an unique form of weight 1000 uh which looks like q to the whatever 1000 over 12 minus four plus high order terms so it has four zeros which lie here and then you increase k you see you will see okay this online this is not a picture of what I'm saying you will see four four zeros and what they do is they start creeping towards the cusp but at the rate which is logarithmic in k so the the result is that there are four zeros I know in a sympathetic for the four zeros the imaginary part looks like log k and the real parts are determined in terms of the zeros of the truncated exponential so here is a truncated exponential it's a polynomial of degree d d is four let's say in this picture it has four inverse zeros and then the statement is that the zeros of this particular cusp forms are asymptotically given by this formula so the real part is essentially the argument of this inverse zero the exponential function and the imaginary part is a log of the weight so this is a new let's call it universality class so before we had eisenstein series and von karei series which have zeros located which have real zeros located either on the bottom part of the boundary or on the imaginary axis and on the boundary of the fundamental domain uh for hecke cusp forms the zeros were dense the uniform was distributed and for this odd family of examples this is just a special case of of of the theorem you have a completely different behavior you have a finite number of zeros and the zeros you can locate quite accurately and they are none of the they look different than the other examples okay so these are all examples of questions you can ask about holomorphic modular forms it's not that subject by any means so right so the last thing I want to say is that uh usually in the birth that you're supposed to be to give presents right so we prepare the little present for peter there's a special volume of the journal the analysis mathematics which is coming out now in december uh and several people in the audience have contributed papers and I think this would be a nice gift for peter's birthday so happy birthday peter